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Mirrors > Home > MPE Home > Th. List > f1orn | Structured version Visualization version GIF version |
Description: A one-to-one function maps onto its range. (Contributed by NM, 13-Aug-2004.) |
Ref | Expression |
---|---|
f1orn | β’ (πΉ:π΄β1-1-ontoβran πΉ β (πΉ Fn π΄ β§ Fun β‘πΉ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dff1o2 6835 | . 2 β’ (πΉ:π΄β1-1-ontoβran πΉ β (πΉ Fn π΄ β§ Fun β‘πΉ β§ ran πΉ = ran πΉ)) | |
2 | eqid 2732 | . . 3 β’ ran πΉ = ran πΉ | |
3 | df-3an 1089 | . . 3 β’ ((πΉ Fn π΄ β§ Fun β‘πΉ β§ ran πΉ = ran πΉ) β ((πΉ Fn π΄ β§ Fun β‘πΉ) β§ ran πΉ = ran πΉ)) | |
4 | 2, 3 | mpbiran2 708 | . 2 β’ ((πΉ Fn π΄ β§ Fun β‘πΉ β§ ran πΉ = ran πΉ) β (πΉ Fn π΄ β§ Fun β‘πΉ)) |
5 | 1, 4 | bitri 274 | 1 β’ (πΉ:π΄β1-1-ontoβran πΉ β (πΉ Fn π΄ β§ Fun β‘πΉ)) |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 β§ wa 396 β§ w3a 1087 = wceq 1541 β‘ccnv 5674 ran crn 5676 Fun wfun 6534 Fn wfn 6535 β1-1-ontoβwf1o 6539 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-3an 1089 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-v 3476 df-in 3954 df-ss 3964 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 |
This theorem is referenced by: f1f1orn 6841 infdifsn 9648 efopnlem2 26156 cycpmcl 32262 |
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